Constraints on the Equations of State of stiff anisotropic minerals: rutile, and the implications for rutile elastic barometry

2019 ◽  
Vol 83 (03) ◽  
pp. 339-347 ◽  
Author(s):  
Gabriele Zaffiro ◽  
Ross J. Angel ◽  
Matteo Alvaro
Keyword(s):  
Equation Of State ◽  
Equations Of State ◽  
Harmonic Approximation ◽  
Unit Cell ◽  
Elastic Behaviour ◽  
Published Data ◽  
Grüneisen Parameter ◽  
Unit Cell Parameters ◽  
Gruneisen Parameter ◽  
Cell Parameters

AbstractWe present an assessment of the thermo-elastic behaviour of rutile based on X-ray diffraction data and direct elastic measurements available in the literature. The data confirms that the quasi-harmonic approximation is not valid for rutile because rutile exhibits substantial anisotropic thermal pressure, meaning that the unit-cell parameters change significantly along isochors. Simultaneous fitting of both the diffraction and elasticity data yields parameters of KTR0= 205.14(15) GPa, KSR0= 207.30(14) GPa, $K_{TR0}^{\prime} $= 6.9(4) in a 3rd-order Birch-Murnaghan Equation of State for compression, αV0= 2.526(16) × 10–5 K–1, Einstein temperature θE = 328(12) K, Anderson-Grüneisen parameter δT = 7.6(6), with a fixed thermal Grüneisen parameter γ = 1.4 to describe the thermal expansion and variation of bulk modulus with temperature at room pressure. This Equation of State fits all of the available data up to 7.3 GPa at room temperature, and up to 1100 K at room pressure within its uncertainties. We also present a series of formulations and a simple protocol to obtain thermodynamically consistent Equations of State for the volume and the unit-cell parameters for stiff materials, such as rutile. In combination with published data for garnets, the Equation of State for rutile indicates that rutile inclusions trapped inside garnets in metamorphic rocks should exhibit negative residual pressures when measured at room conditions.

The Use of Quartz as an Internal Pressure Standard in High-Pressure Crystallography

1997 ◽  
Vol 30 (4) ◽  
pp. 461-466 ◽  
Author(s):  
R. J. Angel ◽  
D. R. Allan ◽  
R. Miletich ◽  
L. W. Finger
Keyword(s):  
High Pressure ◽  
Equation Of State ◽  
Order Equation ◽  
Unit Cell ◽  
Unit Cell Parameters ◽  
Third Order ◽  
Single Crystal Diffraction ◽  
Cell Parameters ◽  
Diamond Anvil ◽  
Pressure Standard

The unit-cell parameters of quartz, SiO2, have been determined by single-crystal diffraction at 22 pressures to a maximum pressure of 8.9 GPa (at room temperature) with an average precision of 1 part in 9000. Pressure was determined by the measurement of the unit-cell volume of CaF2 fluorite included in the diamond-anvil pressure cell. The variation of quartz unit-cell parameters with pressure is described by: a −4.91300 (11) = −0.0468 (2) P + 0.00256 (7) P 2 − 0.000094 (6) P 3, c − 5.40482 (17) = − 0.03851 (2) P + 0.00305 (7) P 2 − 0.000121 (6) P 3, where P is in GPa and the cell parameters are in ångstroms. The volume–pressure data of quartz are described by a Birch–Murnaghan third-order equation of state with parameters V 0 = 112.981 (2) å3, K T0 = 37.12 (9) GPa and K′ = 5.99 (4). Refinement of K′′ in a fourth-order equation of state yielded a value not significantly different from the value implied by the third-order equation. The use of oriented quartz single crystals is proposed as an improved internal pressure standard for high-pressure single-crystal diffraction experiments in diamond-anvil cells. A measurement precision of 1 part in 10 000 in the volume of quartz leads to a precision in pressure measurement of 0.009 GPa at 9 GPa.

TexPat – a program for quantitative analysis of oblique texture electron diffraction patterns

2004 ◽  
Vol 219 (1) ◽  
Author(s):  
Peter Oleynikov ◽  
Sven Hovmöller ◽  
Xiaodong Zou ◽  
Anatoliy P. Zhukhlistov ◽  
Maxim S. Nickolsky ◽  
...  
Keyword(s):  
Automated Detection ◽  
Unit Cell ◽  
Published Data ◽  
Unit Cell Parameters ◽  
Cell Parameters ◽  
Symmetry Properties ◽  
Speed Up ◽  
Diffraction Patterns ◽  
Integrated Intensities

AbstractWe have developed a program – TexPat for quantification of texture patterns in order to facilitate, speed up and improve the accuracy of this analytical method. The program introduces new approaches for automated detection of centre and symmetry axes and simplifies the process of indexing and calculating the unit cell parameters. The main algorithm of the program uses the symmetry properties of the texture pattern images. The successive steps help to process the reflections of the pattern using the peak shape extracted from well-separated peaks. The program generates a list of unit cell parameters, all processed reflections with Miller indices and their integrated intensities. The quality of the results obtained by TexPat is compatible with published data.

scholarly journals Thermoelastic properties of magnesiowüstite, (Mg1−xFex)O: determination of the Anderson–Grüneisen parameter by time-of-flight neutron powder diffraction at simultaneous high pressures and temperatures

2008 ◽  
Vol 41 (5) ◽  
pp. 886-896 ◽  
Author(s):  
Ian G. Wood ◽  
Lidunka Vočadlo ◽  
David P. Dobson ◽  
G. David Price ◽  
A. D. Fortes ◽  
...  
Keyword(s):  
Equation Of State ◽  
Powder Diffraction ◽  
High Pressures ◽  
Equations Of State ◽  
Time Of Flight ◽  
Neutron Powder Diffraction ◽  
Grüneisen Parameter ◽  
Gruneisen Parameter ◽  
Measured State

The ability to perform neutron diffraction studies at simultaneous high pressures and high temperatures is a relatively recent development. The suitability of this technique for determiningP–V–Tequations of state has been investigated by measuring the lattice parameters of Mg1−xFexO (x= 0.2, 0.3, 0.4), in the rangeP < 10.3 GPa and 300 <T< 986 K, by time-of-flight neutron powder diffraction. Pressures were determined using metallic Fe as a marker and temperatures were measured by neutron absorption resonance radiography. Within the resolution of the experiment, no evidence was found for any change in the temperature derivative of the isothermal incompressibility, ∂KT/∂T, with composition. By assuming that the equation-of-state parameters either varied linearly or were invariant with composition, the 60 measured state points were fitted simultaneously to aP–V–T–xequation of state, leading to values of ∂KT/∂T= −0.024 (9) GPa K−1and of the isothermal Anderson–Grüneisen parameter δT= 4.0 (16) at 300 K. Two designs of simultaneous high-P/Tcell were employed during this study. It appears that, by virtue of its extended pressure range, a design using toroidal gaskets is more suitable for equation-of-state studies than is the system described by Le Godec, Dove, Francis, Kohn, Marshall, Pawley, Price, Redfern, Rhodes, Ross, Schofield, Schooneveld, Syfosse, Tucker & Welch [Mineral. Mag.(2001),65, 737–748].

scholarly journals A self-consistent approach to describe unit-cell-parameter and volume variations with pressure and temperature

2021 ◽  
Vol 54 (6) ◽  
Author(s):  
Ross Angel ◽  
Mattia Mazzucchelli ◽  
Javier Gonzalez-Platas ◽  
Matteo Alvaro
Keyword(s):  
Cell Parameter ◽  
Equations Of State ◽  
Unit Cell ◽  
The Self ◽  
Polynomial Functions ◽  
Unit Cell Parameters ◽  
Cell Parameters ◽  
Self Consistent ◽  
Consistent Manner ◽  
Consistent Approach

A method for the self-consistent description of the large variations of unit-cell parameters of crystals with pressure and temperature is presented. It employs linearized versions of equations of state (EoSs) together with constraints to ensure internal consistency. The use of polynomial functions to describe the variation of the unit-cell angles in monoclinic and triclinic crystals is compared with the method of deriving them from linearized EoSs for d spacings. The methods have been implemented in the CrysFML Fortran subroutine library. The unit-cell parameters and the compressibility and thermal expansion tensors of crystals can be calculated from the linearized EoSs in an internally consistent manner in a new utility in the EosFit7c program, which is available as freeware at http://www.rossangel.net.

TheP–V–Tequation of state of D2O ice VI determined by neutron powder diffraction in the range 0

2012 ◽  
Vol 45 (3) ◽  
pp. 523-534 ◽  
Author(s):  
A. D. Fortes ◽  
I. G. Wood ◽  
M. G. Tucker ◽  
W. G. Marshall
Keyword(s):  
Equation Of State ◽  
Bulk Modulus ◽  
Powder Diffraction ◽  
Equations Of State ◽  
Brillouin Scattering ◽  
Personal Communication ◽  
Neutron Powder Diffraction ◽  
Unit Cell ◽  
Cell Parameters ◽  
Neutron Powder Diffraction Data

Neutron powder diffraction data have been collected from deuterated ice VI (space groupP42/nmc,Z= 10) at 76 state points in its field of thermodynamic stability above 150 K (approximately 0.6 <P< 2.1 GPa) and in a region of metastable persistence below 150 K (0 <P< 2.6 GPa). The refined unit-cell parameters and unit-cell volume have been fitted with six-parameter functions based upon a Murnaghan equation of state (EoS), with simple polynomial expressions to describe the temperature dependence of the volume (or axis length) and the bulk modulus (or axial incompressibility). ThisP–V–TEoS is compared with earlier empirically derived and computationally obtained equations of state, and with the elastic properties determined by ultrasonic or Brillouin scattering methods. It is found that D2O ice VI is ∼5% stiffer (i.e.it has a larger bulk modulus,K, at a givenPandT) than H2O ice VI, in agreement with unpublishedcij/ρ data for D2O ice VI provided by the Shimizu Laboratory, Gifu University, Japan (S. Sasaki, personal communication). This difference is due entirely to a greater incompressibility parallel to theaaxis than for the protonated isotopologue; the incompressibilities of thecaxis for both H2O and D2O ice VI are very similar. TheP–V–TEoS is used to estimate the volume change and enthalpy of melting. Additional data on the isothermal compression of ices II and V at 240 K (0.2–0.4 and 0.4–0.7 GPa, respectively), and of ice VII at 295–300 K (2–7 GPa), are also reported and compared with earlier literature data.

Thermoelastic properties and crystal structure of CaPtO3post-perovskite from 0 to 9 GPa and from 2 to 973 K

2011 ◽  
Vol 44 (5) ◽  
pp. 999-1016 ◽  
Author(s):  
Alex Lindsay-Scott ◽  
Ian G. Wood ◽  
David P. Dobson ◽  
Lidunka Vočadlo ◽  
John P. Brodholt ◽  
...  
Keyword(s):  
Crystal Structure ◽  
High Pressure ◽  
Equation Of State ◽  
Powder Diffraction ◽  
Ambient Pressure ◽  
Full Range ◽  
Unit Cell ◽  
Similar Amount ◽  
Grüneisen Parameter ◽  
Gruneisen Parameter

ABX3post-perovskite (PPV) phases that are stable (or strongly metastable) at ambient pressure are important as analogues of PPV-MgSiO3, a deep-Earth phase stable only at very high pressure. The thermoelastic and structural properties of orthorhombic PPV-structured CaPtO3have been determined to 9.27 GPa at ambient temperature and from 2 to 973 K at ambient pressure by time-of-flight neutron powder diffraction. The equation-of-state from this high-pressure study is consistent with that found by Lindsay-Scott, Wood, Dobson, Vočadlo, Brodholt, Crichton, Hanfland & Taniguchi [(2010).Phys. Earth Planet. Inter.182, 113–118] using X-ray powder diffraction to 40 GPa. However, the neutron data have also enabled the determination of the crystal structure. Thebaxis is the most compressible and thecaxis the least, with theaandcaxes shortening under pressure by a similar amount. Above 300 K, the volumetric coefficient of thermal expansion, α(T), of CaPtO3can be represented by α(T) =a0+a1(T), witha0= 2.37 (3) × 10−5 K−1anda1= 5.1 (5) × 10−9 K−2. Over the full range of temperature investigated, the unit-cell volume of CaPtO3can be described by a second-order Grüneisen approximation to the zero-pressure equation of state, with the internal energy calculatedviaa Debye model and parameters θD(Debye temperature) = 615 (8) K,V0(unit-cell colume at 0 K) = 227.186 (3) Å3,K′0(first derivative with respect to pressure of the isothermal incompressibilityK0) = 7.9 (8) and (V0K0/γ′) = 3.16 (3) × 10−17 J, where γ′ is a Grüneisen parameter. Combining the present measurements with heat-capacity data gives a thermodynamic Grüneisen parameter γ = 1.16 (1) at 291 K. PPV-CaPtO3, PPV-MgSiO3and PPV-CaIrO3have the same axial incompressibility sequence, κc > κa > κb. However, when heated, CaPtO3shows axial expansion in the form αc > αb > αa, a sequence which is not simply the inverse of the axial incompressibilities. In this respect, CaPtO3differs from both MgSiO3(where the sequence αb > αa > αcis the same as 1/κi) and CaIrO3(where αb > αc > αa). Thus, PPV-CaPtO3and PPV-CaIrO3are better analogues for PPV-MgSiO3in compression than on heating. The behaviour of the unit-cell axes of all three compounds was analysed using a model based on nearest-neighbourB—XandA—Xdistances and angles specifying the geometry and orientation of theBX6octahedra. Under pressure, all contract mainly by reduction in theB—XandA—Xdistances. On heating, MgSiO3expands (at high pressure) mainly by lengthening of the Si—O and Mg—O bonds. In contrast, the expansion of CaPtO3(and possibly also CaIrO3), at atmospheric pressure, arises more from changes in angles than from increased bond distances.

Kinematical and Dynamical CBED Pattern Models: Increasing the Accuracy of the Unit-Cell Parameter Measurements Based on CBED Patterns

1990 ◽  
Vol 48 (2) ◽  
pp. 540-541
Author(s):  
I.N. Yadhikov ◽  
S.K. Maksimov
Keyword(s):  
Reciprocal Lattice ◽  
Unit Cell ◽  
Reciprocal Lattice Vector ◽  
Unit Cell Parameters ◽  
Lattice Vector ◽  
Cell Parameters ◽  
Accuracy Of Measurements ◽  
The Difference ◽  
Image Position ◽  
Convergent Beam

Convergent beam electron diffraction (CBED) is widely used as a microanalysis tool. By the relative position of HOLZ-lines (Higher Order Laue Zone) in CBED-patterns one can determine the unit cell parameters with a high accuracy up to 0.1%. For this purpose, maps of HOLZ-lines are simulated with the help of a computer so that the best matching of maps with experimental CBED-pattern should be reached. In maps, HOLZ-lines are approximated, as a rule, by straight lines. The actual HOLZ-lines, however, are different from the straights. If we decrease accelerating voltage, the difference is increased and, thus, the accuracy of the unit cell parameters determination by the method becomes lower.To improve the accuracy of measurements it is necessary to give up the HOLZ-lines substitution by the straights. According to the kinematical theory a HOLZ-line is merely a fragment of ellipse arc described by the parametric equationwith arc corresponding to change of β parameter from -90° to +90°, wherevector, h - the distance between Laue zones, g - the value of the reciprocal lattice vector, g‖ - the value of the reciprocal lattice vector projection on zero Laue zone.

Sectioned rubisco crystals in TEM

1990 ◽  
Vol 48 (3) ◽  
pp. 664-665
Author(s):  
Gunnel Karlsson ◽  
Jan-Olov Bovin ◽  
Michael Bosma
Keyword(s):  
Plant Cell ◽  
Carbon Fixation ◽  
Unit Cell ◽  
Protein Crystals ◽  
Unit Cell Parameters ◽  
Cell Parameters ◽  
Photosynthetic Carbon Fixation ◽  
Photosynthetic Carbon

RuBisCO (D-ribulose-l,5-biphosphate carboxylase/oxygenase) is the most aboundant enzyme in the plant cell and it catalyses the key carboxylation reaction of photosynthetic carbon fixation, but also the competing oxygenase reaction of photorespiation. In vitro crystallized RuBisCO has been studied earlier but this investigation concerns in vivo existance of RuBisCO crystals in anthers and leaves ofsugarbeets. For the identification of in vivo protein crystals it is important to be able to determinethe unit cell of cytochemically identified crystals in the same image. In order to obtain the best combination of optimal contrast and resolution we have studied different staining and electron accelerating voltages. It is known that embedding and sectioning can cause deformation and obscure the unit cell parameters.

On the accuracy of determining the unit cell parameters of single crystals on modern laboratory diffractometers

2021 ◽  
Vol 62 (5) ◽  
Author(s):  
П.C. Серебренникова ◽  
В.Ю. Комаров ◽  
А.С. Сухих ◽  
С.А. Громилов
Keyword(s):  
Single Crystals ◽  
Unit Cell ◽  
Unit Cell Parameters ◽  
Cell Parameters

Application of the Redlich-Kwong-Soave equation of state to the solubility of water in compressed gases

1984 ◽  
Vol 49 (5) ◽  
pp. 1116-1121
Author(s):  
Josef P. Novák ◽  
Jaroslav Matouš ◽  
Petr Pick ◽  
Jiří Pick
Keyword(s):  
Equation Of State ◽  
Binary Systems ◽  
Equations Of State ◽  
Published Data ◽  
Interaction Coefficients

Published data on the solubility of water in compressed gases were employed for calculating the interaction coefficients kij in the Redlich-Kwong-Soave equations of state for binary systems of water with argon, nitrogen, CO2, N2O, CH4, C2H6, or C2H4. With these coefficients, the estimate of the solubility of water in these gases has been improved by more than one order.

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